Differential Private Statistic Inference In Bradley-Terry Model

Pairwise comparisons of data are very common in our lives,such as various sports competitions,journal rankings and so on.The Bradley-Terry model is a com-mon model for analyzing pairwise comparison data.When the number of individuals to be compared is large,the design of pairwise comparisons is usually sparse.Sparse is measured by the probability pn of whether there is a comparison between two in-dividuals.The closer the probability is to zero,the more sparse it is.Pair comparison data can be abstractly represented by weighted directed graphs,and Bradley-Terry model can sort the vertices of weighted directed graphs.In many practical problems,the degree sequence of network graph model usually contains a lot of sensitive in-formation.In this thesis,we study the statistical inference problem in Bradley-Terry model under the differential privacy criterion.The main contents are as follows:First,since the degree sequence is a sufficient statistic of the Bradley-Terry model,we use the Laplace mechanism to release the degree sequence satisfying the differential privacy.Based on the difference privacy degree sequence,we use the moment method to estimate the unknown parameters in the Bradley-Terry model,and obtain the estimator satisfying the differential privacy criterion.Second,under certain conditions,we prove the consistency of the differen-tial privacy estimator.Specific expressions are as follows:assuming ?*?Rn,(e2??*?? ?min+e6??*??/pn3)2?=o((n/log n)1/2),where ?=(1+(1/n log n)1/23 log n/?),and ? is the privacy parameter;Then as n goes to infinity,with probability approach-ing one,the estimate ? of ? exists and satisfies Further,if the maximum likelihood estimate exists,it must be unique.Thirdly,under given conditions,the asymptotic normality of the obtained d-ifferential privacy estimator is proved,which is described as follows:assuming ?*?Rn,(e2??*?? ?min+e6??*??/pn3)2?=o((n/log n)1/2),and e2??*??/n3pn6=o(n-1/2),A?P?*,where P?.is probability distribution of Bradley-Terry model network graph with the parameter ?*:Then for any fixed k? 1,as n goes to infinity,with probabil-ity approaching one,the first k elements of(B-1)1/2(?-?*)asymptotically obey the multivariate standard normal distribution,where B=diag(1/v11+1/v00+0?2/v002,…,1/vnn+1/v00+n?2/v002),(?),?2=Var(?)=2?/(1-?)2.Further,the theoretical results are verified by numerical simulation.